Strain Gradient Approach Research Articles

Analyzing torsional vibration in restrained functionally graded nanobeams: nonlocal Lam strain gradient approach

Analyzing torsional vibration in restrained functionally graded nanobeams: nonlocal Lam strain gradient approach

Porosity-dependent wave propagation in multi-directional functionally graded nanoplate with nonlinear temperature-dependent characteristics on Kerr-type substrate

The present study investigates the wave propagation characteristics of porous, multi-directional, functionally graded (FG) nanoplates embedded in a Kerr-elastic substrate. A first-order shear deformation model (FSDM) and the nonlocal strain gradient approach (NSGA) are utilized to address this issue. The NSGA is enhanced by considering softening and hardening material effects, leading to improved accuracy in the obtained results. In accordance with the law of mixing, the material properties of the nanoplates exhibit nonlinear temperature-dependent variations along the structural axes. Both even and uneven porosity profiles are considered in this study. The nonclassical governing equations for the proposed structure are derived using Hamilton's approach. The phase velocity and wave frequency are determined through the analytical solution of an eigenvalue problem, which depends on the wave number. The dispersion properties of waves in porous FG nanoplates are examined with respect to various factors, including porosity distributions, wave numbers, nonlocal and strain gradient effects, elastic foundation coefficients, and volume fraction index. The main findings of the study indicate that increasing the volume fraction index in the thickness direction leads to a decrease in the phase velocity and frequency of wave propagation in a porous FG nanoplate for a given wave number. Additionally, regardless of the porosity pattern, an increase in the porosity coefficient results in a decrease in wave frequency and phase velocity.

Mechanical Buckling of Functionally Graded Cylindrical Nanopanels: A Nonlocal Strain Gradient Approach

Mechanical Buckling of Functionally Graded Cylindrical Nanopanels: A Nonlocal Strain Gradient Approach

Nonlocal strain gradient approach for axial vibration analysis of arbitrary restrained nanorod

Nonlocal strain gradient approach for axial vibration analysis of arbitrary restrained nanorod

Size dependent torsional vibration of a restrained single walled carbon nanotube (SWCNT) via nonlocal strain gradient approach

A torsional vibration model of a restrained nano-sized rod in which deformable boundary effect is considered in the context of nonlocal strain gradient theory is proposed in this work. The strain gradient nonlocal theory introduced from Eringen’s nonlocal model contains additional small scale gradient parameter that represent the high-order strain gradient effects besides the Eringen’s size effect parameter depicting interactions of particles. Within the framework of Fourier infinite series and Stokes’ transform, an eigenvalue problem is employed to obtain the torsional vibration frequencies. The validity of the presented work is checked by comparing the calculated numerical results by giving proper values to elastic spring parameters and good agreement is achieved. This proposed model can be effectively applied to practical and non-idealized torsionally vibrating problems involving deformable boundary conditions. The two size parameters influencing on torsional dynamics of restrained nanorods are study in detail, meanwhile, validation and comparisons of present work with other non-classical elastic models, and the relations between rigid and deformable boundary conditions are shown in this paper.

On dynamic modeling of piezomagnetic/flexomagnetic microstructures based on Lord–Shulman thermoelastic model

We study a time-dependent thermoelastic coupling within free vibrations of piezomagnetic (PM) microbeams considering the flexomagnetic (FM) phenomenon. The flexomagneticity relates to a magnetic field with a gradient of strains. Here, we use the generalized thermoelasticity theory of Lord–Shulman to analyze the interaction between elastic deformation and thermal conductivity. The uniform magnetic field is permeated in line with the transverse axis. Using the strain gradient approach, the beam yields microstructural properties. The analytical solving process has been gotten via applying sine Fourier technique on displacements. Graphical illustrations are assigned to shape numerical examples concerning variations in essential physical quantities. It was observed that the flexomagnetic effect could be extraordinary if the thermal conductivity of the material is higher or the thermal relaxation time of the heat source is lesser. This theoretical study will provide the way of starting studies on magneto-thermoelastic small-scale piezo-flexomagnetic structures based on the heat conduction models.

Size-Dependent Vibration Problem of Two Vertically-Aligned Single-Walled Boron Nitride Nanotubes Conveying Fluid in Thermal Environment Via Nonlocal Strain Gradient Shell Model

The free vibration behavior of two fluid-conveying vertically-aligned single-walled boron nitride nanotubes are studied in the present paper via the nonlocal strain gradient piezoelectric theory in conjunction with the first-order shear deformation shell assumption in thermal environments. It is supposed that the two adjacent boron nitride nanotubes are coupled with each other in the context of linear deformation by van der Waals interaction according to Lennard–Jones potential function. To achieve a more accurate modeling for low-scale structures, both hardening and softening effects of materials are considered in the nonlocal strain gradient approach. The motion equations and associated boundary conditions are derived by means of Hamilton’s variational principle, then solved utilizing differential quadrature method. Numerical studies are done to reveal the effect of different boundary conditions, size scale parameters, aspect ratio, inter-tube distance, and temperature change on the variations of dimensionless eigenfrequency and critical flow velocity.

Analytical solutions for vibrations and buckling analysis of laminated composite nanoplates based on third-order theory and strain gradient approach

A nonlocal model based on the strain gradient approach is developed within the framework of the Third-order Shear Deformation Theory (TSDT) for the investigation of the free vibrations and the critical buckling loads of laminated composite nanoplates. The theory is suitable to deal with thick and thin plates since it includes also the First-order Shear Deformation Theory (FSDT) and the Classical Laminated Plate Theory (CLPT). An analytical procedure based on the Navier approach is employed to obtain the solutions, which are discussed highlighting the effects of the strain gradient, as well as the influence of the geometric ratios and mechanical properties, on the results. The paper aims at providing reliable benchmarks for further developments of the topic to be used as references in future comparison tests.

Third-Order Theory for the Bending Analysis of Laminated Thin and Thick Plates Including the Strain Gradient Effect.

The aim of the paper is the development of a third-order theory for laminated composite plates that is able to accurately investigate their bending behavior in terms of displacements and stresses. The starting point is given by the corresponding Reddy’s Third-order Shear Deformation Theory (TSDT). This model is then generalized to consider simultaneously the Classical Laminated Plate Theory (CLPT), as well as the First-order Shear Deformation Theory (FSDT). The constitutive laws are modified according to the principles of the nonlocal strain gradient approach. The fundamental equations are solved analytically by means of the Navier methodology taking into account cross-ply and angle-ply lamination schemes. The numerical applications are presented to highlight the nonlocal effects on static behavior.

Transverse Free Vibration of Axisymmetric Functionally Graded Circular Nanoplates with Radial Loads

The functionally graded circular nanoplate is a commonly seen component in the nano-electromechanical system. It is indispensable to examine free vibration behaviors of such an axisymmetric nanostructure subjected to uniformly distributed loads in the radial direction. Although the vibration engineering and technology have been fully studied at the macro-scale, there are still many unsolved problems at the micro-scale. The present research aims to promote the theoretical characterization of vibration behaviors at micro/nano-scale and further provide a basis for the development of vibration testing technologies. Using the nonlocal strain gradient approach and Mindlin plate theory, we develop the theoretical model describing the transverse free vibration of the axisymmetric functionally graded circular nanoplate. First, by considering the nonlocal strain gradient constitutive relation, we derive the equation of motion via Hamilton’s principle in polar coordinate system. Subsequently, the differential quadrature method is employed to solve the equation of motion numerically. The natural frequencies of circular nanoplates decrease with an increase of a radial compression while increase with an increase of a radial tension. The first-mode natural frequency reduces to zero under a certain radial compression, bringing about the dynamical instability. The natural frequencies are sensitive to the radial compression, and the clamped boundary constraint is more resistant to external loads than the simply supported one. An increase in the nonlocal parameter results in lower natural frequencies, while an increase in the strain gradient characteristic parameter results in higher ones. It is demonstrated that the strain gradient characteristic parameter has a threshold in the present model for functionally graded circular nanoplates. In the circumstance of a lower nonlocal parameter than the strain gradient characteristic parameter, the circular nanoplate shows hardening behaviors. In the circumstance of a greater nonlocal parameter, the circular nanoplate shows softening behaviors. When the two internal characteristic parameters are equal, the stiffness of nanoplates remains unchanged and degenerates into its classical counterpart.

Hygro-thermo-mechanical vibration of two vertically aligned single-walled boron nitride nanotubes conveying fluid

The nonlocal strain gradient theory, when combined with the first-order shear deformation theory, provides many capabilities in size-dependent structures. The aim of the present study is evaluation of the free vibration behavior of two vertically aligned fluid-conveying single-walled boron nitride nanotubes in hygrothermal environments considering slip boundary condition based on Knudsen number. These two adjacent nanotubes are coupled in the context of linear deformation through van der Waals interaction according to Lennard–Jones potential function. Actually, the contribution of the present work, compared with those previously reported, is investigating the simultaneous effect of hygrothermal loading and slip boundary condition on the dynamic behavior of two vertically aligned fluid-conveying single-walled boron nitride nanotubes. As an additional step to achieve a more accurate model of low-scale structures, both hardening and softening effects of materials are taken as important variables in the nonlocal strain gradient approach. To derive the motion equations and associated boundary conditions, Hamilton’s variational principle is used. The equations are then solved with the aid of differential quadrature method. Numerical studies are also performed to depict the effects of a number of parameters such as boundary conditions, size scale, aspect ratio, inter-tube distance, and temperature alteration on the variations of dimensionless eigenfrequency and critical flow velocity.

A Nonlocal Strain Gradient Approach for Out‐of‐Plane Vibration of Axially Moving Functionally Graded Nanoplates in a Hygrothermal Environment

Vibration analyses on axially moving functionally graded nanoplates exposed to hygrothermal environments are presented. The theoretical model of the nanoplate is described via the Kirchhoff plate theory in conjunction with the concept of the physical neutral layer. By employing the nonlocal strain gradient theory, the governing equation of motion is derived based on Hamilton’s principle. The composite beam function method, as well as the complex modal approach, is utilized to obtain the vibration frequencies of axially moving functionally graded nanoplates. Some benchmark results related to the effects of temperature changing, moisture concentration, axial speed, aspect ratio, nonlocal parameter, and the material characteristic scale parameter on the stiffness of axially moving functionally graded nanoplates are obtained. The results reveal that with increasing the nonlocal parameter, gradient index, temperature changing, moisture concentration, and axial speed, the vibration frequencies decrease. The frequencies increase while increasing the material characteristic scale parameter and aspect ratio. Moreover, there is an interaction between the nonlocal parameter and material characteristic scale parameter, influencing and restricting each other.

Higher order nonlocal viscoelastic strain gradient theory for dynamic buckling analysis of carbon nanocones

The dynamic buckling in viscoelastic carbon nanocones (CNCs) under the magnetic and thermal loads is considered in this article. This class of nano-structures has great application in nano-electro-mechanical system (NEMS) such as scanning probe microscopy. Structural damping influences are studied according to approach of Kelvin–Voigt. The viscoelastic medium around the CNC is assumed applying the model of visco-Pasternak. Utilizing the first order shear deformation theory (FSDT), Hamilton's principle, method of energy, the motion equations are determined where the influences of size are captured by higher order nonlocal strain gradient approach. Method of Bolotin and differential quadrature method (DQM) are utilized to solve the equations of motion in order to access the formation of dynamic instability region (DIR). The influences of different factors like, boundary conditions, nonlocal parameters, structural damping, magnetic field, viscoelastic medium, temperature changes along with CNCs semi vertex angle on the DIR are investigated. The outcomes show that with rising strain gradient parameter, the DIR will be occurred at high frequencies. It is furthermore found that the magnetic load has a positive influence on the DIR of the nano-structure.

Computational frame of ligament in situ strain in a full knee model

The biomechanical function of connective tissues in a knee joint is to stabilize the kinematics-kinetics of the joint by augmenting its stiffness and limiting excessive coupled motion. The connective tissues are characterized by an in vivo reference configuration (in situ strain) that would significantly contribute to the mechanical response of the knee joint. In this work, a novel iterative method for computing the in situ strain at reference configuration was presented. The framework used an in situ strain gradient approach (deformed reference configuration) and a detailed finite element (FE) model of the knee joint. The effect of the predicted initial configuration on the mechanical response of the joint was then investigated under joint axial compression, passive flexion, and coupled rotations (adduction and internal), and during the stance phase of gait. The inclusion of the reference configuration has a minimal effect on the knee joint mechanics under axial compression, passive flexion, and at two instances (0% and 50%) of the stance phase of gait. However, the presence of the ligaments in situ strains significantly increased the joint stiffness under passive adduction and internal rotations, as well as during the other simulated instances (25%, 75% and 100%) of the stance phase of gait. Also, these parameters substantially altered the local loading state of the ligaments and resulted in better agreement with the literature during joint flexion. Therefore, the proposed computational framework of ligament in situ strain will help to overcome the challenges in considering this crucial biological aspect during knee joint modeling. Besides, the current construct is advantageous for a better understanding of the mechanical behavior of knee ligaments under physiological and pathological states and provide relevant information in the design of reconstructive treatments and artificial grafts.

Strain gradient finite element model for finite deformation theory: size effects and shear bands

In this work, a thermodynamically consistent constitutive formulation for the coupled thermomechanical strain gradient plasticity theory is developed in the context of the finite deformation framework. A corresponding finite element solution is presented to investigate the microstructural features of metallic volumes. The developed model is established based on an extra Helmholtz-type partial differential equation, and the nonlocal quantity is calculated in a coupled method based on the equilibrium conditions. This approach is well known for its computational strength, however, it is also commonly accepted that it cannot capture the size effect phenomenon observed in the micro-/nanoscale experiments during hardening. In order to resolve this issue, a modified strain gradient approach which can capture the size effects under the finite deformation is constructed in this work. The shear problem is then solved to carry out the feasibility study of the developed model on the size effect phenomenon. Lastly, a plane strain problem under uniaxial tensile loading with shear bands is examined to perform the mesh sensitivity tests of the model during softening.

Timoshenko nonlocal strain gradient nanobeams: Variational consistency, exact solutions and carbon nanotube Young moduli

Carbon nanotubes (CNTs) are principal constituents of nanocomposites and nano-systems. CNT size-dependent response assessment is therefore a topic of current interest in Mechanics of Advanced Materials and Structures. CNTs are modeled here by a variationally consistent nonlocal strain gradient approach for Timoshenko nano-beams, extending the treatment in [Int. J. Eng. Science 143 (2019) 73–91] confined to slender structures. Scale effects are described by integral convolutions, conveniently replaced with differential and boundary nonlocal laws. The theoretical predictions, exploited to analytically estimate the reduced Young elastic modulus of CNTs, are validated by molecular dynamics simulations.

On torsion of nonlocal Lam strain gradient FG elastic beams

The nonlocal strain gradient theory of elasticity is the focus of numerous studies in literature. Eringen’s nonlocal integral convolution and Lam’s strain gradient model are unified by a variational methodology which leads to well-posed structural problems of technical interest. The proposed nonlocal Lam strain gradient approach is presented for functionally graded (FG) beams under torsion. Static and dynamic responses are shown to be significantly affected by size effects that are assessed in terms of nonlocal and gradient length parameters. Analytical elastic rotations and natural frequencies are established by making recourse to a simple solution procedure which is based on equivalence between integral convolutions and differential equations supplemented with variationally consistent (but non-standard) nonlocal boundary conditions. Effects of Eringen’s nonlocal parameter and stretch and rotation gradient parameters on the torsional behavior of FG nano-beams are examined and compared with outcomes in literature. The illustrated methodology is able to efficiently model both stiffening and softening torsional responses of modern composite nano-structures by suitably tuning the small-scale parameters.

Dynamic modeling of non-cylindrical curved viscoelastic single-walled carbon nanotubes based on the second gradient theory

This paper is devoted to the theoretical study of the dynamic response of non-cylindrical curved viscoelastic single-walled carbon nanotubes (SWCNTs). The curved nanotubes are largely used in many engineering applications, but it is challenging in understanding mechanically the dynamic response of these curved SWCNTs when considering the influences of the material viscosity. The viscoelastic damping effect on the dynamic response is considered here by using the Kelvin-Voigt viscoelastic model. A modified shear deformation beam theory is here employed to formulate the governing partial differential equations. When the SWCNTs are considered in a small scale model, quantum impacts are important for a correct evaluation of the mechanical response of the nanosystem. This is here investigated by embedding the well-known nonlocal strain gradient approach into the governing equations. The extracted equations are solved by utilizing the Galerkin analytical approach in which the governing partial differential equations are reduced to ordinary differential equations and numerical findings are achieved for various boundary conditions. In order to evaluate the efficiency of the proposed theory, the outcomes in terms of natural frequencies of the vibrating nanotubes are verified with respect to the available literature. It follows a vast systematic study, where several parameters are varied to investigate the influences of geometrical properties involving different polygons of the SWCNTs on the dynamic response.

Nano-scale effects on nonlocal boundary conditions for exact buckling analysis of nano-beams with different end conditions

This paper presents a mathematical approach for applying the nano-scale effects on higher-order nonlocal boundary conditions for exact buckling strains of nano-beams based on a modified nonlocal Timoshenko beam theory (MNTBT) for various end conditions. Researchers usually neglect these higher-order boundary conditions in their analysis. Hence, the strain gradient approach and variational method are implemented in MNTBT for deriving these higher-order boundary conditions and exact closed-form critical buckling strains for different end conditions. On this basis, exact numerical results are presented for buckling analysis of single-walled carbon nanotubes (SWCNTs). It is investigated that nonlocal boundary conditions have the effect of reducing the critical buckling strains. This effect is the most significant for doubly clamped nano-beams and the least significant for cantilever nano-beams. Furthermore, it is shown that presented model based on higher-order nonlocal boundary conditions can capture correctly the length-dependent buckling strains of SWCNTs as compared with the other nonlocal beam theories. Finally, the results are compared with molecular dynamics simulations and the Eringen’s nonlocal coefficient (e0) is calibrated for buckling problems of SWCNTs with good accuracy as compared with literatures.

Wave propagation in strain gradient poroelastic medium with microinertia: closed-form and finite element solutions

This article is about ultrasonic wave propagation in microstructured porous media. The classic Biot’s model is enriched using a strain gradient approach to be able to capture high-order effects when the wavelength approaches the characteristic size of the microstructure. In order to reproduce actual transmission/reflection experiments performed on poroelastic samples, and to validate the choice of the model, the computation of the time domain response is necessary, as it allows for a direct comparison with experimental results. For obtaining the time response, we use two strategies: on the one hand we compute the closed form solution by using the Laplace and Fourier transforms techniques; on the other hand we used a finite element method. The results are presented for a transmission/reflection test performed on a poroelastic sample immersed in water. The effects introduced by the strain gradient terms are visible in the time response and in agreement with experimental observations. The results can be exploited in characterization of mechanical properties of poroelastic media by enhancing the reliability of quantitative ultrasound techniques.